Perhaps the most challenging of all guidance and control problems is that of a tactical air-to-air missile in pursuit of a highly maneuverable target aircraft. The problem presented to this missile may be divided into several parts which include the estimation of target motion, the generation of guidance commands to optimally steer the missile toward target intercept and control of the coupled, nonlinear, multivariable, uncertain dynamics of the air-to-air missile. Each portion of this problem, i.e., estimation, guidance and control, is inherently nonlinear and time varying, and a solution of all three problem parts combines to comprise a complex integrated system.
The traditional approach to homing guidance of such a missile employs the proportional navigation (PRONAV) guidance algorithm. The PRONAV algorithm was developed by C. Yuan at RCA Laboratories during World War II using intuition; see Yuan, C. L., "Homing and Navigation Courses of Automatic Target-Seeking Devices," RCA Laboratories, Princeton N.J., Report PTR-12C, December 1942. The resulting simplistic guidance law states that the commanded linear acceleration of the missile, .alpha..sub.c, is proportional to the line-of-sight (LOS) acceleration rate .sigma.. The appropriate proportionality constant can be divided into the product of the effective navigation rate N times the relative missile to target closing velocity V.sub.c yielding EQU .alpha..sub.c =NV.sub.c.sigma. (1)
Two decades later, the quasi-optimality of the PRONAV algorithm was demonstrated, see Bryson, A. E. and Ho, Y. C., Applied Optimal Control, Blaisdell Publishing Company, Waltham Mass., 1969. The prefix quasi is used to describe PRONAV optimality because of the assumptions required in deriving PRONAV as a solution of a linear-quadratic optimal control problem, see the United States Air Force publication by Riggs, T. L. and Vergez, P. L., "Advanced Air-to-Air Missile Guidance Using Optimal Control and Estimation," AFATL-TR-81-56, Air Force Armament Directorate of Wright Laboratory, Eglin AFB FL, June 1981. These assumptions are as follows:
1. The target has zero acceleration. PA1 2. The missile has perfect response and complete control of its acceleration vector. PA1 3. The missile is launched on a near collision course such that the LOS angles remain small over the entire engagement. PA1 4. The missile has zero acceleration along the LOS vector over all time. PA1 bringing said missile within seeker range of said target; PA1 determining flight path azimuth and elevation rates for said missile using an embodied guidance algorithm having target-related, missile-related, and relative target/missile related state terms; PA1 said algorithm including target acceleration, missile velocity, missile acceleration and line-of-sight rate state terms in each of an azimuth signal channel equation and an elevation signal channel equation; PA1 said algorithm also including azimuth signal channel and elevation signal cross-channel coupling-related terms; PA1 generating new flight path azimuth and elevation rates for said missile in response to successive updating changes in said target acceleration, missile velocity, missile acceleration, line-of-sight rate and cross-channel coupling-related algorithm terms; PA1 said generating step including determining new flight path azimuth and elevation rates for said missile following disturbance of a guidance process by either of target maneuvering and missile maneuvering accelerations; PA1 repeating said generating step at selected intervals until said missile is deemed sufficiently close to said target for effective detonation.
In order to remove the first assumption, an additional term is added to the basic PRONAV algorithm in an attempt to account for target acceleration. The additional term is simply the target's estimated linear acceleration, .alpha..sub.T, multiplied by a gain, g.sub.3. In order to remove the fourth assumption, another term, .alpha..sub.M, is sometimes included which attempts to compensate for the missile's acceleration. The resulting guidance law, known as "augmented PRONAV", appears in its most general form as EQU .alpha..sub.c =NV.sub.c.sigma.+g.sub.3 (t.sub.go).alpha..sub.T +g.sub.4 (t.sub.go).alpha..sub.M (2)
where .alpha..sub.T is target acceleration, .alpha..sub.M is missile acceleration and g.sub.3 and g.sub.4 are functions of t.sub.go, which is the time remaining, or time-to-go, until impact or detonation.
Using equation (2) and the small line-of-sight angle assumption, assumption 3 above, the additional augmented PRONAV relationship EQU .alpha..sub.c =g.sub.1 (t.sub.go)y+g.sub.2 (t.sub.go)y+g.sub.3 (t.sub.go).alpha..sub.T +g.sub.4 (t.sub.go).alpha..sub.M (3)
can be derived, where y is relative position and y is relative velocity, with g.sub.1 =N/t.sub.go.sup.2 and g.sub.2 =N/t.sub.go.
Over the past twenty-five years, numerous linear-quadratic optimal control algorithms have been posed attempting to improve upon the augmented PRONAV of equations (2) and (3) and to determine "optimal" values for the gains g.sub.1, g.sub.2, g.sub.3 and g.sub.4 (see for example Lin, C. V., Modem Navigation, Guidance, and Control Processing, Prentice Hall, Englewood Cliffs N.J., 1991, Chapter 8). These linear quadratic formulations have typically been based on Cartesian-based target motion models and notably the resulting guidance law solutions all require knowledge of the time-to-go quantity.
There are two disadvantages associated with the resulting guidance law or algorithm. The first disadvantage is that the states in a Cartesian-based target motion model are nonlinearly related to seeker measurements, which are spherical-based quantities such as range, range rate, and azimuth and elevation angles. Thus, there is a certain amount of incompatibility between seeker measurements and the target motion model. The second disadvantage is the requirement to estimate time-to-go, i.e., t.sub.go. A consistently accurate estimate of t.sub.go cannot be obtained in a maneuvering target scenario since it depends upon the target's future motion which is unknown.
In order to make the target state estimator more compatible with seeker measurements and overcome the first disadvantage, a spherical-based nonlinear intercept kinematics model has been developed by the present inventor and several colleagues. See the publication by D'Souza, C. N., McClure, M. A., and Cloutier, J. R., "Spherical Target State Estimators," Proceedings of the American Control Conference, Baltimore Md., June 1994. Moreover in the above referenced U.S. Pat. No. 6,064,332 the need for t.sub.go estimation was eliminated in the development of guidance laws known as "proportional guidance" (PROGUIDE) and "augmented proportional guidance" (Augmented PROGUIDE). However, these PROGUIDE guidance laws are not based on nonlinear intercept kinematics and do not command flight path angle rate or linear acceleration. Instead, they are based on a simple linear model of the intercept and command flight path angle acceleration.
Subsequently, and as is disclosed in the above referenced and copending patent (S.I.R) application Ser. No. 08/753,754, filed Nov. 29, 1996, and in the technical paper by the present inventor, i.e., the paper of Cloutier, J. R., "Adaptive Matched Augmented Proportional Navigation," presented at the AIAA Missile Sciences Conference, Monterey Calif., November 1994, two improved "time-to-go-less guidance laws" (i.e., guidance algorithms free of the time-to-go parameter estimate) were developed. These subsequent algorithms are based on the nonlinear spherical-based intercept kinematics model disclosed in the above D'Souza, C. N., McClure, M. A., and Cloutier, J. R. publication. Furthermore, it has been demonstrated in six degrees-of-freedom simulations that these guidance laws yield superior performance over even the augmented proportional navigation algorithm. However, these subsequent guidance laws do not account for the cross-channel couplings that exist between the azimuth and elevation guidance channels. Such accounting is however considered in the present invention. The present invention therefore provides additional improvement over the augmented PRONAV guidance algorithm.